Adaptive multistep model predictive control for tubular grid-connected solid oxide fuel cells

: In the research of renewable energy power generation, tubular grid-connected solid oxide fuel cells with the apparent advantage in voltage regulation have been widely applied in power systems. Recently, a model predictive control has been applied to consider the nonlinear constraints of tubular grid-connected solid oxide fuel cells, which cannot be considered by a proportional-integral-derivative controller. While both model predictive control and proportional-integral-derivative controller achieve only 80% fuel efficiency, which should be improved. An adaptive multistep model predictive control (AMMPC) is proposed to improve the fuel efficiency of tubular grid-connected solid oxide fuel cells and simultaneously consider systemic thermodynamics and electrochemistry constraints. The AMMPC contains the advantages of adaptive control and multistep model predictive control. Both adaptive two-step model predictive control and three-step model predictive control are designed for tubular grid-connected solid oxide fuel cells. With the more accurate prediction ability, the AMMPC improves the fuel efficiency of tubular grid-connected solid oxide fuel cells with higher fuel efficiency (86.5%) than model predictive control (80%) and proportional-integral-derivative (80%). Both feasibility and effectiveness of the AMMPC are verified with high fuel efficiency under both simple and complex power demands cases.


Introduction
Recently, renewable energy sources have been introduced to large-scale interconnected power systems for reducing frequency deviation and voltage deviation [1]. For example, tubular grid-connected solid oxide fuel cells (TGSOFCs) have been applied for the voltage regulation of power systems [2]. The combination of electrolytic hydrogen production and hydrogen storage might be an effective way to improve the efficiency of renewable sources [3]. As an auxiliary technology of renewable energy storage, the power system with TGSOFCs has been regarded as a candidate power supply for the voltage regulation of power systems [4].
The TGSOFC with high energy conversion efficiency, fast power response, and stable output voltage has been widely applied in industrial applications [5]. power transmission of a simplified TGSOFC (including an anode, cathode, voltage source inverter, transformer, load, power grid) [6]. Hydrogen is provided to the anode through the fuel processor; oxygen is supplied to the cathode through the air compressor. Individually, the chemical reaction occurs in the electrolyte and generates an electromotive force [6]. The TGSOFCs has the following features: (i) TGSOFCs for peak shaving should rapidly respond to the demand sides of power systems [7]; (ii) the fuel efficiency of TGSOFCs should be configured as the maximum value as possible [8]; (iii) the control system of TGSOFCs is a real-time strongly nonlinear control system [9].
The power system with TGSOFCs has been optimized by a setpoint scheduler and a controlled tracking controller [10,11]. The nonlinear control system for the TGSOFCs is considered in this paper.
The fuel efficiency of TGSOFCs should be configured as much larger as possible to reduce the economic cost of fuel cells [12]. However, the fuel efficiency of TGSOFCs that larger than 90% lead to insufficient fuel for TGSOFCs [13]. The fuel efficiency should be effectively improved in the range from 70% to 90% when considering the safe operation and the economic cost of TGSOFCs [14]. An optimized proportional-integralderivative (PID) can only achieve a maximum of 80% fuel efficiency [15]. An intelligent controller should be designed to improve the fuel efficiency of TGSOFCs.
The model predictive control (MPC) has recently been applied to obtain high fuel efficiency of TGSOFCs than other control methods [16]. Besides, the MPC has been introduced to the tracking controller to improve the fuel efficiency of TGSOFCs [17,18]. The active disturbance rejection control and MPC have been combined with maintaining the fuel efficiency of TGSOFCs to an expected constant [19]. The PID could not be suitable for the constraints of the nonlinear TGSOFCs [20]. The maximum power point tracking algorithms could not effectively improve the fuel efficiency of the TGSOFC with nonlinear constraints [21]. To improve the fuel efficiency near to 90% and provide a control strategy for nonlinear dynamic systems, a more intelligent control method with more accurate prediction should be designed for TGSOFCs. This paper adopts an adaptive multistep model predictive control (AMMPC) for the tracking controller of TGSOFCs.
The AMMPC is proposed to improve the fuel efficiency of a TGSOFC. The AMMPC consists of adaptive control and multistep model predictive control. The AMMPC aims to control TGSOFCs with nonlinearities and constraint characteristics. The AMMPC consists of an adaptive control and multistep model predictive control. Besides, the AMMPC can simultaneously supply multiple output actions. Consequently, the significant contributions of the tracking controller based on the AMMPC for the TGSOFCs are listed as follows: (i) the adaptive control of AMMPC can control TGSOFCs with nonlinearity; (ii) the AMMPC shows the convenience and robustness for nonlinear system constraints; (iii) the AMMPC can gain a stable output voltage and higher fuel efficiency for TGSOFCs. 3 The remaining works of this article are arranged as follows. The TGSOFC is introduced in Section 2.
The control structure of the AMMPC and fuel system protection control are established in Section 3. The AMMPC with higher fuel efficiency and better tracking performances are represented as Section 4.
Conclusions are briefly listed as Section 5.

Tubular grid-connected solid oxide fuel cell models
The principle of TGSOFCs is shown in following subsection.

Framework of tubular grid-connected solid oxide fuel cell models
TGSOFCs have higher airtightness and structural integrity than other types of GSOFC. A TGSOFC includes a cathode, an electrolyte, and an anode (Fig. 2 Since the initial preheating process of TGSOFCs is not high [22], the preheating process can significantly influence its long-term dynamics [23]. Thus, the thermal characteristics of the fuel cell model should be considered. The internal resistance of TGSOFCs increases with the increasing process of the temperature of TGSOFCs and the voltage output would be affected. Based on the comprehensive study of the thermal characteristics of TGSOFCs, Wang and Nehrir proposed a new fuel cell model, which considers the electrochemical and thermal characteristics of TGSOFCs [24]. The TGSOFCs, which have enough accuracy and reasonable complexity, are suitable for modern power systems [24].

Mass conservation of tubular grid-connected solid oxide fuel cells
Assume that: a TGSOFC adopts one-dimensional analysis processing; the stoichiometric proportion of cathode air is larger than that of the anode fuel; the pressure of the air and hydrogen of the TGSOFC is equal everywhere in the air supply tube. Then, the dynamic TGSOFC model and the analysis methods have been built [23]. The pressure of hydrogen with Faraday Law is described as [25] The hydrogen conservation based on Faraday Law is described as [26], The mass conservation of steam based on Faraday Law is described as [26], The conservation of gas mass inside of the air supply channels is described as [27]: The conservation of gas mass at the air supply channels inlet and outlet are displayed as [27], The mass of oxygen at the inlet of air supply channels; and the mass of oxygen of air supply channels outlet are displayed as [27], anode cathode is the total pressure of gas entering anode; out cathode C p is the total pressure of gas living anode.
The fuel efficiency of a TGSOFC is defined as [13]: where  is the fuel efficiency of the TGSOFC.

Thermal equilibrium of tubular grid-connected solid oxide fuel cells
The electrochemical reaction of TGSOFCs, which occurs in the layer of the electrode-electrolyte, can generate heat. The electric energy generated by the electrochemical reaction is transferred to the systemic power load. The generated heat of TGSOFCs is dissipated between the electrodes and the electrolyte. In the air supply channel inlet of TGSOFCs, a cooling device is installed into the layer between electrodes and electrolyte to cool the fuel and air flows. Simultaneously, a small part of the generated heat by TGSOFCs diverts to the air supply channel by thermal transmission. The internal cross-section and the thermal transmission process of the chemical reaction of the TGSOFC are shown in Fig. 3.
The air temperature of the air supply tube of TGSOFCs is equal to the average of the air temperature at the channel inlet and the annulus air temperature of the air supply tube, as [28]  In the electrode-electrolyte layer of the TGSOFC, the heat generated by the electrochemical reaction of the TGSOFC is transferred by three parts, i.e., the radiation consumed of the TGSOFC, the heat loss caused by the conversion between air supply tube and annulus of the TGSOFC, and the heat loss caused by fuel transfer between the annulus and the supply tube of the TGSOFC [28]: out chemical radiation cell,annulus,convection cell,fuel,convection C where chemical Q is the heat generated by the chemical reactions of TGSOFC; out C V is the output voltage of TGSOFC; radiation Q is the radiation consumed of TGSOFC; cell,annulus,convection Q is the heat loss caused by the conversion between the air supply tube and the annulus of TGSOFC; cell,fuel,convection Q is the heat loss caused by fuel transfer between the annulus and the supply tube of TGSOFC. 7 Eq. (16) includes the heat consumed by the internal resistance of the TGSOFC and the thermal transmission ways of the TGSOFC.
The heat generated by the fuel combustion of the TGSOFC can be described as [29]  The heat loss caused by fuel transfer between the annulus and the supply tube of the TGSOFC can be calculated as [29]: The heat transfer between the annulus and the air supply tube of the TGSOFC can be calculated as [30]: outer outer cell air supply tube,annulus,convection air supply tube, air supply tube, air, air supply tube

Electric energy transfer in the chemical reaction
The open-circuit potential of TGSOFC obeys Nernst Equation, as [28]:  The concentration voltage drop after the chemical reaction of the TGSOFC can be calculated as [28]: The inner voltage drop of the TGSOFC can be described as [28]:

Adaptive multistep model predictive control
Adaptive control and multiple MPCs stack the proposed AMMPC.

Model predictive control
The MPC structure includes a model, a predictor, and a controller (Fig. 4). 10

Control Predictor
Model h  ,t x  +  is the increment of the variable  of the  iteration in the control horizon.
The following constraints should be satisfied for the manipulated variables to obtain the increments [33]. The adaptive control law is applied to update the estimation ˆ() ij s  , which is presented as [34]: where  is the adaptation rate of the controlled objective; Projection q expresses the projection operator of controlled systems, which guarantees the estimation value is bounded [35].
The predicted outputs of the AMMPC are described as,

If step
N is set to be two, the AMMPC with two steps is named AMMPC-II; if step N is set to be three, the AMMPC with three steps is named AMMPC-III. Each weighted coefficient of AMMPC should be configured for real-life projects.

Flow chart of adaptive multistep model predictive control for tubular grid-connected solid oxide fuel cells
A multiple-output controller with two outputs can be presented as out TT 12 [ , ] [ , ] Then, the input hydrogen and oxygen are expressed as After the setpoints scheduler of the TGSOFC provides the voltage value for the AMMPC, the AMMPC provides the hydrogen for the TGSOFC. Meanwhile, to satisfy the operation safety of TGSOFC, the fuel efficiency of the TGSOFC system should be maintained between 70% and 90% [14]. The AMMPC method is designed to improve fuel efficiency by regulating the fuel flow and airflow of TGSOFC.     Two cases (i.e., simple and complex power demand cases) are studied to verify the feasibility and effectiveness of the AMMPC. In this paper, four control algorithms are compared under power systems with a TGSOFC. The compared control algorithms include the PID, the adaptive MPC, the AMMPC-II, and the AMMPC-III.
The AMMPC-III and AMMPC-II can provide the output voltage and fuel efficiency by manipulating the inlet air and hydrogen flow of the TGSOFC. Since one PID controller only provides one output, two controllers are designed to control the TGSOFC (Fig. 7). One PID controller is utilized to provide the reference current for the VSI by feedback output power. Another PID controller is employed to provide output voltage 14 by dominating hydrogen flow and airflow. Then, the fuel efficiency is feedback to the reference fuel efficiency for the next hydrogen flow regulation.   Table 1 and Fig. 8.   -III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC-III  AMMPC- The optimal fuel efficiency of the AMMPC-III controller is that when the weight of the first controller is 0.9 and the weight of the second and third controllers are both 0.05 ( Fig. 8 and Table 1). The fuel efficiencies of the TGSOFC obtained by the AMMPC-II with constraint and different weights of two predicted controllers under Case I are shown in Table 2 and Fig. 9.
Table2. Fuel efficiencies obtained by adaptive two-step model predictive control (Case I)  Fig. 9. Fuel efficiency curves obtained by adaptive two-step model predictive control (Case I) As shown in Table 2 and Fig. 9, the optimal control effect of the AMMPC-II controller is that when the weight of the first controller is 0.9 and the weight of the second controller is 0.1, the fuel efficiency of the 16 TGSOFC is the best in steady-state and transient process. The AMMPC-II algorithm can improve the fuel efficiency of the TGSOFC to 86.4%, and the fuel efficiency based on the MPC controller is 80.0%. This paper compares the optimal weight controller of the AMMPC-III control algorithm with the optimal weight controller of the AMMPC-II control algorithm. The output power curves of the TGSOFCs with the proposed AMMPC-III and three compared methods (AMMPC-II, MPC, PID) are shown in Fig. 10. The TGSOFCs with AMMPC-III controller obtain higher fuel efficiency.
With the simple structure of the PID, the thermal characteristics of the resistance considered have little influence on the output power. With a complex control system of the MPC tracking controller, the internal resistance of TGSOFCs is increased; thus, the output powers of the MPC and AMMPCs are lower than that of the PID. Fig. 10 shows that (i) with the increasing internal resistance consumption, the AMMPC can provide the output power at a stable constant; (ii) the AMMPC-III (86.5%) achieves higher control performance than the AMMPC-II (86.4%).  Fig. 11 shows that both AMMPC-III and AMMPC-II can positively improve the fuel efficiency larger than 86%. Besides, the proposed AMMPC-III can obtain higher tracking performance than AMMPC-II, MPC, and PID.  The control performances obtained by four compared methods under Case I are given in Table 3, which shows that the AMMPC can obtain the highest control performance than other compared methods. 19

Case of complex power demand (Case II)
More complex power demand is designed Case II. In this case, the output power curves of the TGSOFCs obtained by compared methods are shown in Fig. 14. The fuel efficiency curves of four compared control methods under the complex power demand are presented in Fig. 15. The output voltage and current curves of four compared control methods under the complex power demand are given in Fig. 16.  can be solved by the AMMPC; (iii) with more next steps are predicted, the AMMPC with three-step can obtain 22 higher fuel efficiency during the changes of power demand than the AMMPC with two-step ( Fig. 14 and Fig.   15); (iv) with more air is used by the TGSOFC, lesser hydrogen is needed for the TGSOFC; thus, both AMMPC and MPC need lesser cost and obtain higher fuel efficiency (Fig. 17).
After numerous testing: (i) the AMMPC with four steps has lower higher fuel efficiency than the AMMPC-III and AMMPC-II; (ii) the AMMPC-II and AMMPC-III with the weight of 0.9 of the first steps controller can obtain higher fuel efficiency than the AMMPC with other weights. The possible reason is that: the control system of the TGSOFC is a real-time short-term time-scale control system. Therefore, more steps (larger than three) could not be needed for the AMMPC-based TGSOFC. For a long-term time-scale control system, (i) the steps of the AMMPC should be configured as one of the hyperparameters; (ii) the number of the steps of the AMMPC for a long-term time-scale control system can be started from two.

Conclusions
This paper presents an AMMPC, which contains adaptive control and multistep MPC. The proposed AMMPC can predict the next systemic states with more accuracy than the MPC. Two case studies (i.e., simple and complex power demand cases) under power systems with TGSOFCs show that: the AMMPC can achieve the optimal value of current, can reduce the power loss caused by internal resistance, and can increase the fuel efficiency of TGSOFCs. The significant features of the AMMPC can be summarized as follows: (1) The AMMPC contains the adaptive control and the multistep model predictive control, updating the control strategy online. The adaptive control of the AMMPC can adjust automatically according to system operation with the optimal control conditions. The multistep model predictive control can predict systemic state more accuracy with more steps is introduced into the AMMPC.
(2) The AMMPC can obtain higher fuel efficiency (86%) than the MPC (80%) and the PID (80%) for TGSOFCs. The AMMPC-III that contains three steps can obtain higher fuel efficiency than the AMMPC-II that contains two steps, which can obtain higher fuel efficiency than the MPC that contains only one step.
(3) The AMMPC can provide a dynamic control strategy for complex nonlinear systems with dynamic characteristics. More next steps information can be applied to a control algorithm.
In the future works, (i) the regulation economic performance of the TGSOFCs could be considered; (